PSI - Issue 13

A.R. Torabi et al. / Procedia Structural Integrity 13 (2018) 596–600 Torabi et al./ Structural Integrity Procedia 00 (2018) 000 – 000

3

598

where K I is the stress intensity factor (SIF) related to a crack of length a stemming from the notch tip, and σ x is the tensile stress along the axial loading (Fig. 1). For positive geometries the strain energy release rate function (and thus the SIF K I ) is monotonically increasing with a : at incipient failure Eq. (2) will turn into a system of two equations in two unknowns: critical crack advance  c and nominal failure stress σ f , which is implicitly embedded in functions K I and σ x . Table 1. Experimental failure loads and average failure stresses. PMMA GPPS 2 R (mm) P f (N) f  (MPa) 2 R (mm) P f (N) f  (MPa) 0.5 37150 28.377 0.5 15930 15.020 0.5 33810 0.5 15280 0.5 36020 0.5 14090 1 28800 22.003 1 12260 11.986 1 26230 1 11970 1 27920 1 11920 2 21830 18.469 2 9950 9.3106 2 24700 2 9130 2 23100 2 9000 4 20250 15.453 4 8070 7.7190 4 19936 4 7645 4 18074 4 7565 8 14414 12.205 8 6525 6.2767 8 16168 8 6220 8 15432 8 6185 For sufficiently small hole radii, the SIF function related to a crack of length a stemming from the notch can be approximated with the one valid for a circular hole in an infinite plate, where ( ) ( ) I K a a F s    (3) with The shape function F related to the symmetric crack propagation (Fig. 1a) can be written as (Tada et al. 2000):     3 2 ( ) (1 ) 0.5 (3 ) [1 1.243(1 ) ] 1 (1 ) [0.5 0.743(1 ) ] F s s s s s             (5) where  = ‒3 according t o the present geometry. Furthermore, Kirsch solution (note that in the centre of a plain specimen the stress is equal to  3  in the y -direction) yields the approximate solution for the stress field function: s a  a R  (4)

 

  

2 R R

4



( ) y

2 2 12

   

x 

(6)

2

4

y

y

2

Equation (6) provides ideally a stress concentration factor σ x ( y=R )/ σ = 6, meaning that the maximum stress is 6 times higher than the nominal one. Indeed, the accuracy of Eqs. (3) and (6) could be improved for not negligible R / R 0 ratios by taking the following multiplying factor into account:

2

0 R         1 19 3 R

F

(7)

corr

The FFM criterion (1) can now be implemented by means of Eqs. (3), (6) and (7). By introducing the function f related to the integration of the stress field (6) and the function g related to the integration of the SIF squared (3), some analytical manipulations allow to write the FFM system at failure as / c c R    :